Hereditary quasi-random properties of hypergraphs

Thomason and Chung, Graham and Wilson were the first to investigate systematically some properties of quasi-random graphs. They have stated several quite disparate properties of random-like graphs and established their equivalence. vits and Sós introduced a new hereditary property that is equivalent to the other quasi-random properties. For a small fixed graph F, a graph G on n vertices is said to have the Simonovits-Sós Property SSP if for every set X ⊆ V ( G ) , the number of labeled copies of F in G [ X ] is given by 2 − e ( F ) | X | v ( F ) + o ( n v ( F ) ) . A graph that satisfies SSP for some non-empty graph F is quasi-random. ntribution in this paper is a natural extension of the result of Simonovits and Sós to 3-uniform hypergraphs.